When converting numbers from one base to another, it is important to remember that in a positional system, we group quantities into the largest place value possible. For example, in the number 234 in base ten, we have 2 hundreds rather than 20 tens.

An analogy you can use is packaging. Imagine that you are trying to package a quantity of items in the most efficient way, using the least number of boxes (groups). Thus, in base ten you would fill all the boxes that can hold 100 before you start filling the boxes that hold only 10; you'd fill the size 10 boxes before the boxes that hold only one and so on.

The same idea applies to base two. Your packages, however, are of different sizes (32, 16, 8, ...). So, when converting between bases, we are essentially repackaging from using one set of boxes (...1,000, 100, 10, 1, ... for base ten) to another (...32, 16, 8, 2, 1, ...for base two). Note that this process works both ways and for all bases.

So, to convert or repackage a base ten integer such as 52 to base two, first you determine which boxes (groups) would be the most efficient, and then you use the least number of boxes.

Let's work it out step by step:

52ten = _____two

Step 1: Record the base ten powers of 2 from right to left, starting with 20, or 1. Continue until you reach the place where the next power would be greater than the base ten number you are trying to convert. Using the packaging analogy, these will be your new boxes for repackaging.

Here are the base ten powers of 2:

25

24

23

22

21

20

32

16

8

4

2

1

In our case, we can stop at 25 since 26 = 64, and 64 is greater than 52. This box would be too large for 52 items.

Step 2: Next, record a 1 in the place of the greatest power less than your number (i.e., the biggest box you can use), and subtract that base ten value from your number.

For example, we can fill one size 32 box from the 52 items, so write a 1 in that place:

32

16

8

4

2

1

1

Subtract to see what's left:

52 - 32 = 20

Step 3: Now look at the difference. What is the next biggest power of 2 (the next-size box) we can use?

In this case, 20 is greater than the next-smaller power of 2 (i.e., 16) so we can fill one box of 16 items as well. Write a 1 in that place:

32

16

8

4

2

1

1

1

Subtract to see what's left:

20 - 16 = 4

Step 4: Continue filling the remaining boxes until the remainder is 0, recording a 1 or a 0 as required.

For example, you couldn't fill a box of size 8 with four items, so write a 0 in that place. But you could fill the next-smaller box with four items. Write a 1 in that place:

32

16

8

4

2

1

1

1

0

1

Subtract to see what's left:

4 - 4 = 0

Since we have reached 0, we know we are done. As a result, there is no 2 and no 1, and we can write a 0 in each of those places. We have just efficiently repackaged the number 52, using the fewest new boxes (powers of two):

Video SegmentIn this video segment, Doug and Tom convert numbers from base two to base ten and vice versa. Doug notices that the number 63 in base ten is one less than the next power of two, and he explains how his finding is reflected in the answer. Watch this segment after you've completed Problems A2 and A3.

Can you apply the packaging analogy to explain Doug's finding?

If you are using a VCR, you can find this segment on the session video approximately 9 minutes and 54 seconds after the Annenberg Media logo.